A simple formulation for designing fixed order dynamic compensators which are robust to both uncertainty at the plant input and structured uncertainty in the plant dynamics is presented. The emphasis is on designing low order compensators for systems of high order. The formulation is done in an output feedback setting which exploits an observer canonical form to represent the compensator dynamics. The formulation also precludes the use of direct feedback of the plant output. The main contribution lies in defining a method for penalizing the states of the plant and of the compensator, and for choosing the distribution on initial conditions so that the loop transfer matrix approximates that of a full state design. To improve robustness to parameter uncertainty, the formulation avoids the introduction of sensitivity states, which has led to complex formulations in earlier studies where only structured uncertainty has been considered

Byrns, Edward V., Jr.

Calise, Anthony J.

English

NASA Technical Reports Server

N90-23059
ROBUST FIXED ORDER DYNAMIC COMPENSATION
FOR LARGE SPACE STRUCTURE CONTROL
Anthony J. Calise and Edward V. Byrns, Jr.
Georgia Institute of Technology
School of Aerospace Engineering
Atlanta, GA 30332
ABSTRACT
This paper presents a simple formulation for designing fixed order
dynamic compensators which are robust to both uncertainty at the plant
input and structured uncertainty in the plant dynamics. The emphasis is
on designing low order compensators for systems of high order. The
formulation is done in an output feedback setting which exploits an
observer canonical form to represent the compensator dynamics. The
formulation also precludes the use of direct feedback of the plant output.
The main contribution lies in defining a method for penalizing the states
of the plant and of the compensator, and for choosing the distribution on
initial conditions so that the loop transfer matrix approximates that of a
full state design. To improve robustness to parameter uncertainty, the
formulation avoids the introduction of sensitivity states, which has led to
complex formulations in earlier studies where only structured uncertainty
has been considered.
INTRODUCTION
Linear Quadratic Regulator (LQR) design methods are easy to use and
have guaranteed stability margins at the plant input. Unfortunately, this
requires full state feedback for implementation. With Loop Transfer
Recovery (LTR) techniques, the loop characteristics of an LQR design for
square, minimum phase plants can be asymptotically realized using output
feedback with a full order observer. 1 This design methodology can be used
to improve the robustness of observer based controller designs to
unstructured uncertainty. However, for control of large order plants, this
approach may result in controllers that are computationally expensive to
implement. Moreover, although there is good robustness to unstructured
uncertainty at the plant input, the design may remain sensitive to
structured uncertainty in the plant parameters.
Optimal output feedback of fixed order dynamic compensators 2 has
received limited attention due to numerous difficulties associated with
this approach. Initially, the proposed compensator representation was
overparameterized, which means it lacked a predefined structure. To
https://ntrs.nasa.gov/search.jsp?R=19900013743 2020-03-19T22:15:55+00:00Z
overcome this obstacle, several canonical structures have been proposed
which result in a minimal parameterization. 3,4 This study utilizes the
observer canonical form since it yields a convenient form when the design
is treated as a constant gain output feedback problem.
Another major objection to fixed order dynamic compensation is
that there are no guarantees on the stability margins. This drawback was
eliminated by a new methodology for designing fixed order compensators, s
This technique approximates the LQR/LTR method, by appropriate
selection of the plant and compensator state weighting matrices. Much
like the full order observer design, a two step process is used. First, full
state gains are computed for desirable loop properties, followed by the
approximate LTR compensator design. The fundamental assumption of this
approach is that if the closed loop return signals of the two designs are
equal, then the loop transfer functions (with the loop broken at the plant
input) must be equivalent. Unlike the LQR/LTR design, there is no
requirement that the system be minimum phase or square.
A popular method of parameter sensitivity reduction consists of
including a quadratic trajectory sensitivity term in the linear regulator
formulation. In Ref. 6, the approximate LTR methodology for low order
compensators was extended to include sensitivity reduction for real plant
parameter variations. The sensitivity reduction is accomplished by a
simple modification of the quadratic performance index. Unlike earlier
studies in this area, 7,8,9 this formulation does not require increasing the
dimension of the problem to include the sensitivity states. In addition,
the parameter sensitivity reduction is achieved with minimal sacrifice in
the loop transfer characteristics.
An outline of this paper is as follows. First, the approximate LTR
methodology of Ref. 5 is reviewed. Then, the sensitivity reduction
formulation is presented for the spec_c case of a scalar uncertainty in
the state equation. The generalizations of the trajectory sensitivity
approach, given in Ref. 6, are summarized afterwards. Several future
extensions of this research are also discussed.
CONTROLLER DESIGN FORMULATIONS
Dynamic Compensation
Consider a linear system Of the form
_p = Apxp +Bpup XpE: 9_n, UpE 9_rn
yp = _Xp + Dpup ypE 9_P
(1)
(2)
where yp represents the measured outputs. A dynamic compensator can be
parameterized in the following observer canonical form: 4
_, -- P°z + u ZOF_.c_nc (3)
U = PzUp - Nyp UoE:C_nc (4)
Up = - H°z (5)
where nc is the order of the compensator and Pz and N are the matrices
containing the design parameters. The matrices po and H ° are specified by
the choice of observability indices, and their structures are given in Ref.
4.
An equivalent, constant gain output feedback problem is obtained
using the augmented system dynamics.
}_ = Ax + Bu c (6)
y=Cx (7)
uc = -Gy (8)
where x T = {Xp T, zT}, yT ___{ypT, .upT} and
Ap -BpH °-
A=
0 pO B=EoIInc
OF
Cp - DpH °"
0 H°
G=IN Pz_ (10)
The performance index in this case is
,; xo{folXTOx•uT.u °t} (11)
643
It is shown in Ref. 5 that the loop transfer properties of a full state
feedback design are approximately recovered at the plant input by
choosing the following weighting matrices in (11).
Q=[ K*TK* -K*TH° 1 R= pBTB pinc
k .HOTK * HOTH o J
(12)
(13)
where K* is the full state feedback gain matrix, and repeating the design
for decreasing values of p. Although the full state loop transfer
properties are approximately recovered as p is reduced, the controller
design may be sensitive to parameter uncertainty.
Uncertain State Equation
We first consider the system (1,2) with a scalar uncertainty
parameter ec in the state equation.
_p = Ap((X)Xp + Bp((X)Up (14)
In the case of dynamic compensation, (5) becomes
= A((x)x + BUc (15)
where
A(o0 =
Ap(=) - Bp(=)H °
0 pO
(16)
The trajectory sensitivity dynamics are obtained by differentiating
(13) with respect to (z.
= Aczx + _ -BGC)a; G(0) = 0 (1 7)
644
where o = onx/ono_lo_o, A s = onA(o_)/ono_lo_o, _, = A(o, o) and o_o denotes the
nominal value for o_. The standard approach would consider minimizing the
following performance index.
,. xo/I: xTox.uc .uo,o  IotI ,'
where the weighting matrix S is used to penalize sensitivity to parameter
uncertainty. However, this increases the dimension of the problem to
2(n+nc). To avoid this drawback, we adopt the following viewpoint. Note
that Ae_x acts as a forcing function in (15), and that (_(0) = 0. Also, note
that since G stabilized the nominal system, the the dynamics of c(t) are
also stable. Thus, Iloll can be reduced by penalizing IIA xII. This suggests
that the follSwing index of performance be used to introduce a penalty on
sensitivity to parameter uncertainty
J= Exo/Io [xT(Q +_ATAo_)x + uTRuct dt (19)
Thus, a second design parameter, _, is introduced which can be used
to penalize sensitivity to parameter variations, without increasing the
order of the dynamic system. When Q, R and Xo are chosen in accordance
with (12) and (13), then increasing TI permits a design trade off between
desirable loop transfer properties at the plant input and parameter
sensitivity reduction. Equations (7), (8), (15) and (19), with A(_)= A,
constitute a static optimal output feedback problem, whose necessary
conditions for optimality are well known. 1°
This approach to parameter sensitivity reduction can be generalized
to include an uncertain output equations. With uncertainty in the state
and output equation, the trajectory sensitivity dynamics become
(_= (A(z "BG_a)x + (h, -BG__,)_; o(0) = 0 (20)
where the input matrix B is specified by the observer canonical structure
in (9). To minimize a(t), the logical extension is to penalize II(A¢-
BGC¢)xll in the performance index. Since this penalty depends explicitly
on the gain matrix G, the standard necessary conditions for optimal output
feedback no longer apply. In Ref. 6, these new necessary conditions are
given as well as a generalization of this methodology to include a vector
645
parameters. Also, the design approach is illustrated by a high bandwidth
controller for a flexible satellite.
EXTENSIONS
There are several directions in which the current research on low
order compensation is heading. First, the approximate LTR methodology
used at the plant input has been extended to recovery of loop properties at
the plant output. This design technique uses both the controller and the
observer canonical forms and the duality which exists between these
structures. Similar to the dual LTR formulation of Ref. 1, a full order
observer is first designed for desirable loop properties at the dual system
input. With the appropriate quadratic state and control penalties, the
compensator design approximately recovers these loop characteristics at
the plant output. This formulation can easily be extended to include a
structured uncertainty penalty similar to the trajectory sensitivity
dynamics used in this paper.
The next extension of this work is to develop a design methodology
which will allow simultaneous approximate LTR at both the plant input
and the plant output. This will allow the extension to H-Infinity design.
Specifically, using the design approach in Ref. 11, the H-Infinity controller
becomes a constant gain full state feedback design. This can be viewed as
a design approach that simultaneously achieves loop shaping with the loop
broken at both the plant input and the plant output. Thus, the loop
characteristics of a H-Infinity design can be realized with a low order
dynamic compensator if simultaneous LTR can be achieved.
SUMMARY
A method has been developed for designing fixed order dynamic
compensators which are robust to both unstructured uncertainty at the
plant input and structured uncertainty in the plant dynamics. The design
approach specifies weighting matrices which allow the loop transfer
properties to approximate that of a full state design. The robustness to
real structured parameter variations is accomplished by a modification of
the quadratic performance index. The approach avoids the introduction of
the sensitivity states. Hence, it does not increase the dimension of the
problem to acheive the robustness to real plant parameter variations.
Extension to H-Infinity design are currently under development.
646
REFERENCES
• Doyle, J. C. and Stein, G., "Multivariable Feedback Design: Concepts for
a Classical/Modern Synthesis," IEEE Trans. on Automatic Control,
Vol. AC-26, Feb. 1981, pp. 4-16.
1 Johnson, T. L. and Athans, M., "On the Design of Optimal Constrained
Dynamic Compensators for Linear Constant Systems," IEEE Trans. on
Automatic Control, Vol. AC-15, Dec. 1970, pp. 37-41.
. Martin, G. D. and Bryson, A. E., "Attitude control of a Flexible
Spacecraft," Journal of Guidance and Control, Vol. 3, Jan.-Feb. 1980,
pp. 37-41.
. Kramer, F. S. and Calise, A. J., "Fixed Order Dynamic Compensation for
Multivariable Linear Systems," Journal of Guidance, Control and
Dynamics, Vol. 11, Jan.-Feb. 1988, pp. 80-85.
o Calise, A. J. and Prasad, J. V. R., "An Approximate Loop Transfer
Recovery Method for Designing Fixed Order Compensators," AIAA
Guidance, Navigation and Control Conf., Paper No. 88-4078,
Minneapolis, MN, Aug. 1988. (To appear in AIAA J. of Guidance,
Control and Dynamics).
, Calise, A. J. and Byrns, E. V. Jr., "Robust Fixed Order Dynamic
Compensation," AIAA Guidance, Navigation and Control Conf., Paper
No. 88-4078, Boston, MA, Aug. 1989.
1 Kriendler, E., "On the Definition and Application of the Sensitivity
Function," Journal of the Franklin Institute, Vol. 285, pp. 26-36,
1968.
. O'Reilly, J., "Low Sensitivity Feedback Controllers for Linear Systems
with Incomplete State Information," International Journal of Control,
Vol. 29, No. 6, 1979, pp. 1042-1058.
. Okada, K. and Skelton, R., "A Sensitivity Controller for Uncertain
Systems," AIAA Guidance, Navigation and Control Conf., Paper No.
88-4077, Minneapolis, MN, Aug. 1988.
10. Mendel, J. M., "A Concise Derivation of Optimal Constant Limited State
Feedback Gains," IEEE Trans. on Automatic Control, Vol. AC-19, Aug.
1974, pp. 447-448.
647
11. Petersen, I. R., "Disturbance Attenuation and H-Infinity Optimization:
A Design Method Based on the Algebraic Riccati Equation," IEEE
Trans. on Automatic Control, Vol. AC-32, May 1987, pp. 427-429.
: _77
648


Robust fixed order dynamic compensation for large space structure control

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server